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Properties

Label 1-2600-2600.1011-r1-0-0
Degree $1$
Conductor $2600$
Sign $0.0964 + 0.995i$
Analytic cond. $279.408$
Root an. cond. $279.408$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.913 − 0.406i)3-s + (−0.5 − 0.866i)7-s + (0.669 − 0.743i)9-s + (−0.669 − 0.743i)11-s + (0.913 + 0.406i)17-s + (−0.913 − 0.406i)19-s + (−0.809 − 0.587i)21-s + (−0.669 − 0.743i)23-s + (0.309 − 0.951i)27-s + (−0.913 + 0.406i)29-s + (−0.809 + 0.587i)31-s + (−0.913 − 0.406i)33-s + (−0.978 − 0.207i)37-s + (0.978 + 0.207i)41-s + (−0.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (−0.5 − 0.866i)7-s + (0.669 − 0.743i)9-s + (−0.669 − 0.743i)11-s + (0.913 + 0.406i)17-s + (−0.913 − 0.406i)19-s + (−0.809 − 0.587i)21-s + (−0.669 − 0.743i)23-s + (0.309 − 0.951i)27-s + (−0.913 + 0.406i)29-s + (−0.809 + 0.587i)31-s + (−0.913 − 0.406i)33-s + (−0.978 − 0.207i)37-s + (0.978 + 0.207i)41-s + (−0.5 − 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0964 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0964 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.0964 + 0.995i$
Analytic conductor: \(279.408\)
Root analytic conductor: \(279.408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (1:\ ),\ 0.0964 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2350913882 + 0.2134119334i\)
\(L(\frac12)\) \(\approx\) \(0.2350913882 + 0.2134119334i\)
\(L(1)\) \(\approx\) \(1.049061290 - 0.3635194486i\)
\(L(1)\) \(\approx\) \(1.049061290 - 0.3635194486i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (0.978 + 0.207i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.16620730302477844637897154551, −18.43996540726156452235501450886, −17.81051080531677616787238340580, −16.61277007508843907957081371548, −16.14749827591655635216489951677, −15.26630052125438818112861666583, −14.96217382996473075384183919879, −14.15671037799440440665780732287, −13.24785644029153338103666537444, −12.71689126144383638107213758244, −11.99970653197760703458956880161, −10.97776282359059061492989397820, −10.07311221034601046151334287538, −9.592866394132270909100539137969, −8.98319225627984796254735890568, −7.94419530590314235028533369938, −7.6442420551366290983487293103, −6.49952615894156841991395196203, −5.562345853515875881396942301798, −4.892949129801404602007337585649, −3.84507664512085900723037142504, −3.22396430091819526792903584594, −2.257512790889870730407024883508, −1.77392949736416170511087915171, −0.04762249364987078277846186270, 0.84769981804244324371767661059, 1.85845701608820251780935478770, 2.767684719698482655248507046348, 3.58635739528590979708641070574, 4.08501328288478721931574569638, 5.32634525002655193080346937528, 6.22958865894673958557329183336, 7.04371373684891197071384755720, 7.627705789309524544501824594259, 8.47171279169767907073428798413, 8.97632313340096830343040243649, 10.13776782435371788499119137030, 10.40976465578155537147876858510, 11.42296113106515919168750861010, 12.642872901585453436330337556387, 12.8469004855517014660637728216, 13.71047570683757417785202634026, 14.27928436071098008372676884662, 14.932738091798037973140113210762, 15.85441990095247677292132231725, 16.47388837799048877212601942998, 17.18566189942739847275333828179, 18.1783871401343855827083758230, 18.75242994703987828326283737760, 19.39836570420088371637364538087

Graph of the $Z$-function along the critical line